Number Theory

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Now in its second edition, this textbook offers an introduction to number theory through the lens of prime number density and properties. It provides a solid foundation in standard number theory topics while also presenting an overview of the entire discipline. Essential subjects include the fundamental theorem of arithmetic, congruences, quadratic reciprocity, arithmetic functions, and prime distribution. This edition introduces p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods for primality testing. Key features include a thorough introduction to analytic number theory, complete proofs of Dirichlet's Theorem and the Prime Number Theorem, and a concise overview of algebraic number theory, covering primes and unique factorization in algebraic number fields. The AKS algorithm is discussed, demonstrating that primality testing can be performed in polynomial time, a topic often omitted in similar texts. Additional topics such as cryptography, Fermat and Mersenne numbers, and Carmichael numbers are also explored. The approachable style, historical context, and a variety of exercises—ranging from simple to challenging—make this textbook suitable for both self-study and classroom settings. A basic understanding of calculus, multivariable calculus, and linear algebra is all that is required, with necessary concepts from abstract algebra and complex analysis introduced as needed.